Short-term eschatological expectations came and went almost cyclically throughout the church history, with or without the prediction of specific dates; there is good evidence that round figures such as AD 500 or 1000 were considered as deadlines in some circles; Joachim of Fiore specifically pointed to 1260 etc.
Perhaps the earliest, pre-Christian Biblical example of setting and revising dates for the end is provided by the consecutive additions in the conclusion of Daniel (1290 days, 1335 days, 12:11f). But this is probably only the tip of the iceberg, as obviously failed prophecies tend to be actively forgotten if they are not successfully reinterpreted.
As to the 1st-century, there is massive evidence of short-term expectations for the parousia -- Matthew 10:23, 24:34//, 1 Thessalonians 4 ("we the living") to name but a few. No memory of specific dates, although warnings like "nobody knows the day or hour" (Matthew 24:36//), or "It is not for you to know the times or periods that the Father has set by his own authority," (Acts 1:7) are a strong indication that chronological "prophecies" were indeed frequent in the early Christian nebula (see also 2 Thessalonians 2:1f, "As to the coming of our Lord Jesus Christ and our being gathered together to him, we beg you, brothers and sisters, not to be quickly shaken in mind or alarmed, either by spirit or by word or by letter, as though from us, to the effect that the day of the Lord is already here").